Knots with Hopf crossing number at most one

We consider diagrams of links in $S^2$ obtained by projection from $S^3$ with the Hopf map and the minimal crossing number for such diagrams. Knots admitting diagrams with at most one crossing are classified. Some properties of these knots are exhibited. In particular, we establish which of these knots are algebraic and, for such knots, give an answer to a problem posed by Fiedler.